Abstract:

Introduction

Previous experimental results have shown that worked-example learning can promote the solution of near, but not far, transfer problems. However, according to Sweller, in order to promote the solution of the far transfer problem, it was necessary to learn a series of worked-examples of variant problems solutions. Furthermore, they must try to solve problems requiring variant rules. Thus, they will be assigned a large number of homework exercises. To avoid this, we developed a rule worked-example learning method to promote far transfer problem solving, in which students applied rules variant reasoning after prototype worked-example learning. We carried out four experiments to test the effectiveness of this method.

Experiment 1

162 fourth-grade students were selected as participants. They were randomly divided into three groups. After learning the prototype worked-examples, the first group learned worked-examples of the four variant problem solutions. The second group applied rule variant reasoning to four problems presented to them. The third group solved four near transfer problems. Then, participants in all groups were evaluated by transfer tests.

The results showed that the far transfer scores of the first group were significantly better than those of the second group (p < 0.001, 95% CI = [0.57, 1.76]) and the third group (p < 0.001, 95% CI = [2.20, 3.09]), and that the second group’s scores were significantly better than those of the third group (p < 0.001, 95% CI = [0.95, 2.02]) (see Figure 1).

Experiment 2

54 mathematics high-performing students, 54 mathematics middle-performing students, and 54 mathematics low-performing students were selected as participants. After learning the prototype worked-examples, they all applied rule variant reasoning to four variant problems presented to them. Then, they all took transfer tests.

The results showed that significant differences were found in the far transfer test scores among three math performance levels, F(2, 161) = 149.74, p < 0.001, η² = 0.65. The far transfer test scores of high-performing students were significantly better than middle-performing students (p < 0.001, 95% CI = [1.06, 1.90]) and low-performing students (p < 0.001, 95% CI = [2.36, 2.93]); The far transfer test scores of middle-performing students were significantly better than low-performing students (p < 0.001, 95% CI = [0.76, 1.57]) (see Figure 2).

Experiment 3

90 mathematics middle-performing students were randomly divided into three groups. Additionally, 90 mathematics low-performing students were randomly divided into three groups. After prototype worked-examples learning, two first groups made up the variant problems by self, and then they carried out rules variant reasoning for the variant problems; two second groups carried out rules variant reasoning for four variant problems presented to them; two third groups made four types division for eight variant problems presented to them, and then they carried out rules variant reasoning for the four kinds of the variant problems. Finally, they were all tested by transfer tests.

The results showed that the significant differences were found in the far transfer test scores among three groups of math middle-performing students, F(2, 179) = 24.16, p < 0.001, η_p² = 0.22; but there were not significant differences in the far transfer test scores among three groups of math low-performing students, F(2, 179) = 1.28, p> 0.05 (see Figure 3).

Experiment 4

80 mathematics low-performing students were randomly divided into two groups. After learning prototype worked-examples, they all made four types division for eight variant problems presented to them. The first group carried out rules variant reasoning using the variant problems of incomplete solving rules. The second group carried out rules variant reasoning for the four kinds of the variant problems. Finally, they all took transfer tests.

The results showed that the far transfer test scores of the first group were significantly better than those of the second, F(1, 79) = 180.78, p < 0.001, η_p² = 0.70 (see Figure 4).

Conclusion

The rule variant reasoning after learning prototype worked-example significantly promotes far transfer problem solving.

Key words: rule worked-example learning, variant problems, rule variant reasoning, variant rules, far transfer problems